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About 1,106 papers for “Sublinear function”

Variable Exponent Function Spaces related to a Sublinear Expectation

Bochi Xu · Abstract and Applied Analysis · 2020

In this paper, variable exponent function spaces<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">L</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced separators="" open="(" close=")"><mml:mrow><mml:mo>·</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:msup></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">L</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>b</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced separators="" open="(" close=")"><mml:mrow><mml:mo>·</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:msubsup></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">L</mml:mi></mml:mrow><mml:mrow><mml:mrow><mml:mi>c</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mfenced separators="" open="(" close=")"><mml:mrow><mml:mo>·</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:mrow></mml:msubsup></mml:math>are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.

Cited by 2Open access

Strong type inequalities for sublinear operator on weighted modular Banach function spaces

Rovshan Bandaliyev, K. K. Omarova, Bahar S. Jafarova · Filomat · 2025

In this paper, we prove the boundedness of sublinear operator on weighted modular Banach function space (BFS) under a certain size condition. We establish sufficient conditions on weight functions and on the geometry of modular BFS for the validity of the strong inequality for sublinear operator on weighted modular BFS under certain size condition. We will assume that the BFS is p-convex and the modular defining the BFS satisfies some growth condition. In particular, we obtain the boundedness of the sublinear operator on weighted Musielak-Orlicz spaces. The size condition is satisfied by most of the operators in harmonic analysis, such as the Calder?n-Zygmund singular integral operator, Hardy-Littlewood maximal operator, Bochner-Riesz means at the critical index, Carleson maximal operator, Ricci-Stein?s oscillatory singular integrals, C. Fefferman?s singular multiplier operator, R. Fefferman?s singular integral operator and so on. The main result is new in the case of an unweighted setting.

Cited by 0Open access

Modulation of neuronal input-output function by subthreshold electric fields from dendritic sublinear integration

Yaqin Fan, Xile Wei, Meili Lu et al. · 2019

Electrical field (EF) is a popular tool for both basic research and clinical applications, its actions on neuronal activities have been investigated from physiological mechanism and dynamics. However, few studies explore its modulatory influence on neuronal computation from the point of view of dendritic sublinear integration caused by passive dendrites which play an important role in neuronal computation. Here with a reduced biophysical model this problem is explained by observing the impact of EF on neuronal computation and dendritic sublinear operation. It is found that the positive EF results in more linear dendritic sublinear integration because of hyperpolarization in distal dendrites together resulting in higher neuronal excitability in neuronal computation but negative EF inhibits this ability due to more pronounced dendritic sublinear operation resulting from the hyperpolarization in distal dendrites. Further, we explain the modulation of positive EF on dysfunctional neuron combining with Feature Binding Problem. This work builds the gap between neuronal computation and dendritic sublinear operation, which is helpful to understand the modulation of EFs on brain functions.

Cited by 1

Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets

Shadi Haddad, Abhishek Halder · arXiv (Cornell University) · 2022

We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.

Cited by 0Open access

Nonlinear Extensions of Farkas’ Lemma with Applications to Global Optimization and Least Squares

V. Jeyakumar, B. M. Glover · Mathematics of Operations Research · 1995

A nonlinear extension of Farkas’ lemma for systems involving the difference of sublinear functions is presented. This extension of Farkas’ lemma is applied to give a complete characterization of global optimality for constrained global optimization problems in which the objective function is the difference of a convex and sublinear function and the constraints are systems of difference sublinear functions. An application to certain fractional programming problems is also given. These results are achieved with merely a stronger consistency condition which reduces to the usual feasibility requirement for problems with sublinear constraints. Further generalizations of Farkas’ lemma for systems involving convex and difference sublinear functions are also presented. Moreover, a generalized Farkas’ lemma for certain specially structural convex inequality systems is shown to be related to the solution of appropriate constrained least squares problems.

Cited by 25

Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets

Shadi Haddad, Abhishek Halder · 2023

We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sub-linear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.

Cited by 0

Sublinear biLipschitz equivalence and sublinearly Morse boundaries

Gabriel Pallier, Yulan Qing · arXiv (Cornell University) · 2022

A sublinear biLipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function $κ$, $κ$-Morse boundaries are defined for all geodesic proper metric spaces as a quasi-isometrically invariant and metrizable topological space of quasi-geodesic rays. In this paper, we prove that $κ$-Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi-geodesic rays. As an application we distinguish a pair of right-angled Coxeter groups brought up by Behrstock up to sublinear biLipschitz equivalence. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.

Cited by 0Open access

Sublinear bilipschitz equivalence and sublinearly Morse boundaries

Gabriel Pallier, Yulan Qing · Journal of the London Mathematical Society · 2024

Abstract A sublinear bilipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function, the associated sublinearly Morse boundaries are defined for all geodesic proper metric spaces as a quasi‐isometrically invariant and metrizable topological space of quasi‐geodesic rays. In this paper, we prove that sublinearly‐Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi‐geodesic rays. As an application, we distinguish a pair of right‐angled Coxeter groups brought up by Behrstock up to SBE. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays.

Cited by 1

On the Time-Varying Constraints and Bandit Feedback of Online Convex Optimization

Xuanyu Cao, K. J. Ray Liu · 2018

In this paper, online convex optimization (OCO) problem with time-varying constraints is studied from the perspective of an agent taking sequential actions. Both the objective function and the constraint functions are dynamic and unknown a priori to the agent. We first consider the scenario of function feedback, in which complete information about the objective function and constraint functions is revealed to the agent after an action is submitted. We propose a computationally efficient online algorithm, which only involves direct closed-form computations at each time instant. It is shown that the algorithm possesses sublinear regret with respect to the dynamic benchmark sequence and sublinear constraint violations, as long as the drift of the benchmark sequence is sublinear, or in other words, the underlying dynamic optimization problems do not vary too drastically. Furthermore, we investigate the scenario of bandit feedback, in which, after an action is chosen, only the values of the objective function and the constraint functions at several random points close to the action are announced to the agent. A bandit version of online algorithm is proposed and we also establish its sublinear expected regret and sublinear expected constraint violations. Finally, two numerical examples, namely online quadratic programming and online logistic regression, are presented to corroborate the effectiveness of the proposed algorithms and to confirm the theoretical guarantees.

Cited by 4

Online Convex Optimization With Time-Varying Constraints and Bandit Feedback

Xuanyu Cao, K. J. Ray Liu · IEEE Transactions on Automatic Control · 2018

In this paper, online convex optimization problem with time-varying constraints is studied from the perspective of an agent taking sequential actions. Both the objective function and the constraint functions are dynamic and unknown a priori to the agent. We first consider the scenario of the gradient feedback, in which, the values and gradients of the objective function and constraint functions at the chosen action are revealed after an action is submitted. We propose a computationally efficient online algorithm, which only involves direct closed-form computations at each time instant. It is shown that the algorithm possesses sublinear regret with respect to the dynamic benchmark sequence and sublinear constraint violations, as long as the drift of the benchmark sequence is sublinear, or in other words, the underlying dynamic optimization problems do not vary too drastically. Furthermore, we investigate the scenario of the bandit feedback, in which, after an action is chosen, only the values of the objective function and the constraint functions at several random points close to the action are announced to the agent. A bandit version of the online algorithm is proposed and we also establish its sublinear expected regret and sublinear expected constraint violations under the assumption that the drift of the benchmark sequence is sublinear. Finally, two numerical examples, namely online quadratic programming and online logistic regression, are presented to corroborate the effectiveness of the proposed algorithms and to confirm the theoretical guarantees.

Cited by 93

DIFFERENTIALLY PRIVATE SUBLINEAR ALGORITHMS

Mukherjee, Tamalika · Figshare · 2023

Collecting user data is crucial for advancing machine learning, social science, and government policies, but the privacy of the users whose data is being collected is a growing concern. {\em Differential Privacy (DP)} has emerged as the most standard notion for privacy protection with robust mathematical guarantees. Analyzing such massive amounts of data in a privacy-preserving manner motivates the need to study differentially-private algorithms that are also super-efficient. <br> This thesis initiates a systematic study of differentially-private sublinear-time and sublinear-space algorithms. The contributions of this thesis are two-fold. First, we design some of the first differentially private sublinear algorithms for many fundamental problems. Second, we develop general DP techniques for designing differentially-private sublinear algorithms. <br> We give the first DP sublinear algorithm for clustering by generalizing a subsampling framework from the non-DP sublinear-time literature. We give the first DP sublinear algorithm for estimating the maximum matching size. Our DP sublinear algorithm for estimating the average degree of the graph achieves a better approximation than previous works. We give the first DP algorithm for releasing $L_2$-heavy hitters in the sliding window model and a pure $L_1$-heavy hitter algorithm in the same model, which improves upon previous works. <br> We develop general techniques that address the challenges of designing sublinear DP algorithms. First, we introduce the concept of Coupled Global Sensitivity (CGS). Intuitively, the CGS of a randomized algorithm generalizes the classical notion of global sensitivity of a function, by considering a coupling of the random coins of the algorithm when run on neighboring inputs. We show that one can achieve pure DP by adding Laplace noise proportional to the CGS of an algorithm. Second, we give a black box DP transformation for a specific class of approximation algorithms. We show that such algorithms can be made differentially private without sacrificing accuracy, as long as the function has small global sensitivity. In particular, this transformation gives rise to sublinear DP algorithms for many problems, including triangle counting, the weight of the minimum spanning tree, and norm estimation.

Cited by 0Open access

DIFFERENTIALLY PRIVATE SUBLINEAR ALGORITHMS

Mukherjee, Tamalika · Purdue · 2023

Collecting user data is crucial for advancing machine learning, social science, and government policies, but the privacy of the users whose data is being collected is a growing concern. {\em Differential Privacy (DP)} has emerged as the most standard notion for privacy protection with robust mathematical guarantees. Analyzing such massive amounts of data in a privacy-preserving manner motivates the need to study differentially-private algorithms that are also super-efficient. <br> This thesis initiates a systematic study of differentially-private sublinear-time and sublinear-space algorithms. The contributions of this thesis are two-fold. First, we design some of the first differentially private sublinear algorithms for many fundamental problems. Second, we develop general DP techniques for designing differentially-private sublinear algorithms. <br> We give the first DP sublinear algorithm for clustering by generalizing a subsampling framework from the non-DP sublinear-time literature. We give the first DP sublinear algorithm for estimating the maximum matching size. Our DP sublinear algorithm for estimating the average degree of the graph achieves a better approximation than previous works. We give the first DP algorithm for releasing $L_2$-heavy hitters in the sliding window model and a pure $L_1$-heavy hitter algorithm in the same model, which improves upon previous works. <br> We develop general techniques that address the challenges of designing sublinear DP algorithms. First, we introduce the concept of Coupled Global Sensitivity (CGS). Intuitively, the CGS of a randomized algorithm generalizes the classical notion of global sensitivity of a function, by considering a coupling of the random coins of the algorithm when run on neighboring inputs. We show that one can achieve pure DP by adding Laplace noise proportional to the CGS of an algorithm. Second, we give a black box DP transformation for a specific class of approximation algorithms. We show that such algorithms can be made differentially private without sacrificing accuracy, as long as the function has small global sensitivity. In particular, this transformation gives rise to sublinear DP algorithms for many problems, including triangle counting, the weight of the minimum spanning tree, and norm estimation.

Cited by 0Open access

POINT DERIVATIONS FOR LIPSCHITZ FUNCTIONS AND CLARKE’S GENERALIZED DERIVATIVE

V. F. Demyanov (st. Petersburg · 2015

Abstract. Clarke’s generalized derivative f0(x, v) is studied as a function on the Banach algebra Lip(X, d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed x ∈ X and fixed v ∈ E the function f0(x, v) is continuous and sublinear in f ∈ Lip(X, d). It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz’s product rule and are thus point deriva-tions. A characterization of the support set in terms of point derivations is given.

Cited by 0

On the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Relation and the Subdifferential Operator of a Sublinear Function

Heinz H. Bauschke, Xianfu Wang, Liangjin Yao · Contemporary mathematics - American Mathematical Society · 2012

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximal monotone operators provided that Rockafellar’s constraint qualification holds. In this note, we provide a new maximal monotonicity result for the sum of a maximal monotone linear relation and the subdifferential operator of a proper, lower semicontinuous, sublinear function. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results on the Fitzpatrick function.

Cited by 13

Markov selections and Feller properties of nonlinear diffusions

David Criens, Lars Niemann · Stochastic Processes and their Applications · 2024

In this paper we study a family of nonlinear (conditional) expectations that can be understood as a diffusion with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function. We establish its Feller properties and examine how to linearize the associated sublinear Markovian semigroup. In particular, we observe a novel smoothing effect of sublinear semigroups in frameworks which carry enough randomness. Furthermore, we link the value function corresponding to the semigroup to a nonlinear Kolmogorov equation. This provides a connection to the so-called Nisio semigroup.

Cited by 4Open access

Sublinear preferential attachment combined with the growing number of\n choices

Yury Malyshkin · arXiv (Cornell University) · 2019

We prove almost sure convergence of the maximum degree in an evolving graph\nmodel combining a growing number of local choices with sublinear preferential\nattachment. At each step in the growth of the graph, a new vertex is\nintroduced. Then we draw a random number of edges from it to existing vertices,\nchosen independently by the following rule. For each edge, we consider a sample\nof the growing size of vertices chosen with probabilities proportional to the\nsublinear function of their degrees. Then new vertex attaches to the vertex\nwith the highest degree from the sample. Depending on the growth rate of the\nsample and the sublinear function, the maximum degree could be of the sublinear\norder, of the linear order or having almost all edges drawing to it. The prove\nusing various stochastic approximation processes and a large deviation\napproach.\n

Cited by 0Open access

A general form for precise asymptotics for complete convergence under sublinear expectation

Xue Ding · AIMS Mathematics · 2021

&lt;abstract&gt;&lt;p&gt;Let $ \{X_n, n\geq 1\} $ be a sequence of independent and identically distributed random variables in a sublinear expectation $ (\Omega, \mathcal H, {\mathbb {\widehat{E}}}) $ with a capacity $ {\mathbb V} $ under $ {\mathbb {\widehat{E}}} $. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b34"&gt;34&lt;/xref&gt;]&lt;/sup&gt;.&lt;/p&gt;&lt;/abstract&gt;

Cited by 14Open access